Exercise 8, § 1

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module. We say that $M$ satisfies the property $(S_k)$ if one has $\mathrm{prof}_{A_p}(M_p) \geq \inf(k, \dim_{A_p}(M_p))$ for every prime ideal $p$ of $A$. a) Every module satisfies $(S_k)$ for $k \leq 0$; saying that a module satisfies $(S_1)$ means that it has no embedded associated prime ideals. b) Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of finitely generated $A$-modules, and let $k$ be an integer. Suppose that $M$ satisfies $(S_k)$, that $M''$ satisfies $(S_{k-1})$, and if $k \geq 2$ that $\mathrm{Ass}(M'') \subset \mathrm{Ass}(M)$. Prove that $M'$ satisfies $(S_k)$ (use Cor. 2 of No. 7).

c) Let $(L,p)$ be a resolution of $M$ by finitely generated free $A$-modules. If $A$ satisfies the property $(S_k)$, the $A$-module $B_i(L)$ for $i \geq 0$ satisfies $(S_h)$ with $h = \inf(k,i+1)$.

d) If $A$ satisfies $(S_2)$, the dual of every finitely generated $A$-module satisfies $(S_2)$ (apply c).

e) Let $J$ be an ideal of $A$, generated by an $M$-regular sequence $(x_1,\ldots,x_r)$. If $M$ satisfies $(S_k)$, the $A$-module $M/JM$ satisfies $(S_{k-r})$.

f) Let $B$ be a noetherian $A$-algebra, $N$ a finitely generated $B$-module which is a faithfully flat $A$-module, and $k$ an integer. If the $B$-module $M \otimes_A N$ satisfies the property $(S_k)$, the $A$-module $M$ does so analogously. If $M$ satisfies $(S_k)$ and if the $(\kappa(p) \otimes_A B)$-module $\kappa(p) \otimes_A N$ satisfies $(S_k)$ for every $p \in \mathrm{Supp}(M)$, the $B$-module $M \otimes_A N$ satisfies $(S_k)$.